My Math Forum Infinite sets

 Algebra Pre-Algebra and Basic Algebra Math Forum

January 2nd, 2014, 07:06 PM   #1
Senior Member

Joined: Oct 2013
From: Far far away

Posts: 429
Thanks: 18

Infinite sets

For question 36. I tried drawing rays from the center of the circle. This does give a 1-1 correspondence between the points on the circle and the triangle outside. BUT as the rays travel outside to meet the sides of the triangle they diverge. This causes some points on the triangle to be missed out (no 1-1 correspondence).

For question 37. I tried to draw rays from a point inside the triangle. Again we get a 1-1 correspondence as required but due to divergence of the rays some points on the circle miss out.
Attached Images
 Untitled.png (22.6 KB, 80 views)

 January 2nd, 2014, 07:53 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Infinite sets Can you demonstrate a one-to-one correspondence between the points of two line segments of differing lengths?
January 2nd, 2014, 08:19 PM   #3
Senior Member

Joined: Oct 2013
From: Far far away

Posts: 429
Thanks: 18

Re: Infinite sets

Quote:
 Originally Posted by MarkFL Can you demonstrate a one-to-one correspondence between the points of two line segments of differing lengths?
This is my best attempt. Attempting to make a 1-1 correspondence between points on segment DE and segment BC. There is a problem in that the lines from A diverge leaving some points on BC without a partner point on DE. However, if I draw segments perpendicular to BC e.g. segment FG there is a 1-1 correspondence between the line segment BC and BA+AC.
Where am I going wrong?
Attached Images
 Untitled2.png (2.6 KB, 74 views)

 January 2nd, 2014, 08:40 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,931 Thanks: 2205 It's impossible to identify a point on BC without a partner point on DE.
January 2nd, 2014, 09:17 PM   #5
Senior Member

Joined: Oct 2013
From: Far far away

Posts: 429
Thanks: 18

Re:

Quote:
 Originally Posted by skipjack It's impossible to identify a point on BC without a partner point on DE.
What about the diverging lines? The lines from A always cut the segments DE and BC into unequal lengths. How can there be a 1-1 correspondence for ALL the points on the two segments (DE and BC)?

January 2nd, 2014, 11:22 PM   #6
Math Team

Joined: Mar 2012
From: India, West Bengal

Posts: 3,871
Thanks: 86

Math Focus: Number Theory
Re: Infinite sets

Quote:
 Originally Posted by shunya What about the diverging lines?
The divergent lines doesn't miss anything -- it's just that neither of the set of points on both lines are countable. Use the definition of bijectivity here. For any point on BG, join it with A and intersect with DE. You have surjectivity. For injectivity, note that only a single straightline can be formed with 2 points.

January 3rd, 2014, 05:03 AM   #7
Math Team

Joined: Sep 2007

Posts: 2,409
Thanks: 6

Re: Infinite sets

Quote:
 Originally Posted by shunya Please look at the picture and answer the question. For question 36. I tried drawing rays from the center of the circle. This does give a 1-1 correspondence between the points on the circle and the triangle outside. BUT as the rays travel outside to meet the sides of the triangle they diverge. This causes some points on the triangle to be missed out (no 1-1 correspondence). For question 37. I tried to draw rays from a point inside the triangle. Again we get a 1-1 correspondence as required but due to divergence of the rays some points on the circle miss out.
You are mistaking your picture for what really happens- there is a separate ray for each point. Your image of some points of the triangle being "missed out" is incorrect. For every such "missed out" point there is a ray that you have not drawn that crosses it.

January 3rd, 2014, 06:08 AM   #8
Global Moderator

Joined: Dec 2006

Posts: 20,931
Thanks: 2205

Quote:
 Originally Posted by shunya How can there be a 1-1 correspondence for ALL . . .
There is a 1 to 1 correspondence between each integer n and each even integer 2n, yet the set of integers seems twice as large as the set of even integers. However, the point you made is valid for finite sets.

Another example: the arctan function, defined on all the reals, is invertible, so it provides a 1-1 correspondence between all real numbers and the open interval (-?/2, ?/2).

 Tags infinite, sets

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Giddyotwiggy Applied Math 1 June 10th, 2011 02:48 AM conradtsmith Applied Math 1 March 24th, 2010 04:07 AM scoracle Applied Math 4 October 9th, 2009 08:28 AM DoctorHandles Advanced Statistics 2 February 13th, 2009 01:25 PM Infinity Applied Math 15 January 1st, 2007 04:24 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top